- Email: [email protected]
Oceanic electric currents induced by fluid convection
Physics of the Earth and Planetary Interiors, 7 (1973) 137—142 © North-Holland Publishing Company, Amsterdam — Printed in The Netherlands
OCEANIC ELECTRIC CURRENTS INDUCED BY FLUID CONVECTION
*
R.S. PECKOVER UKAEA Cuiham Laboratory, A bingdon (Great Britain)
Accepted for publication January 9, 1973
In addition to the electric potentials induced by the gyral motions in the oceans, horizontal electric fields and currents result from the exchange of water between the depths and the surface in the presence of the main geomagnetic field. In this note a simple model representing such a circulation is considered, and the spatial distribution of the correspondin~induced 2. electric calculated. A surface velocity 2 knots could These fields steadyiscurrents in the ocean could be of comparable with induce the normal electric currents up to 10— Amp/rn oceanic daily variation. Since a proportion of the electric current would return through the earth below the sea floor, this calculation provides an upper limit to this component. An indication is also given of the magnetic field distortion and associated electric currents which occur in a highly conducting (Jovian) ocean.
I. Introduction It is customary to consider the thin-sheet approximation for the oceans and to work with electric field and current distributions which are quantitiesvertically averaged through the thin sheet (Rikitake, 1966; Bullard and Parker, 1971; Bullard, 1972). In this note the effects of a simple form of vertical motion are considered. The form of the vertical distribution is considered for a convective cell in the ocean which consists of a relatively fast flowing stream on top and only such return flow beneath as continuity requires. The integrated effect of a fast flowing stream has been considered by Malkus and Stern (1952), and Glasko and Sveshnikov (1961) have examined a model of two layers of fluid with different velocities, Motions in an electrically conducting fluid permeated by a magnetic field induce electric currents which flow within the field and distort to some extent the magnetic field. The distortion of the main geomagnetic field F is negligible and this is well known (Longuet-Higgins, 1949; Larsen, 1968). However the induced electric *
This paper was presented at the I.A.G.A. Workshop on Electromagnetic Induction, held at the University of Edinburgh, 20—27 September 1972.
field and the resulting currents need not be negligible. Here the spatial distribution of fields is calculated for a simple, rather stylized model, which incorporates some features of a real ocean.
2. Model and equations The model considered here consists of a plane horizontal ocean which is of infinite horizontal cxtent, of uniform depth h metres, isotropic electrical conductivity a, and which is permeated by a vertical magnetic field of strength F. It is bounded above by a non-conducting atmosphere, and below by a nonconducting layer of bottom sediment and crustal rock. A convective circulation is postulated which is in the form of quasi-two dimensional rolls where the axis of the roll (the z-direction) has a length scale large compared with the horizontal convective length scale A (see Fig.l). The effects of sphericity can be neglected since the vertical scale of the ocean is ~-1O~ of the radius of the earth. Although a is obviously a function of temperature and salinity, and oceanic streams have temperatures different from their surrounding water, this dependence is weak, and the assumption of constant conductivity by Longuet-Higgins et al. (1954)
138
-
R.S. Peckover, Oceanic electric currents
~
~
~
~‘
/ / ///,
( / h metres
/ —
~////
/ 7/ / /77/
crustal rocks
o’ o —
/
in the of the Eq.presence 1 and 2 can be magnetic combinedinduction together B. into the
/ /7/.
hydromagnetic induction equation:
plane ocean infinite extent
~B —
o— const.
F
7/ / /777
curl (UAB)
V B
= i~
(3)
1is the resistivity. where (~ia) With~ithese equations, it follows that the electric 4
1V
is followed here. The Coriolis force is ignored since no motions are considered in the z-direction. At British latitudes the horizontal component is less than half the vertical component. In fact in the linearized case the solutions for horizontal and vertical components can easily be superimposed. Here only a vertical component is considered. The electrical conductivity of the atmosphere is a factor of 1012 less than that of the ocean, and the crustal rocks are a factor of 10~less conductive than the ocean (Larsen, 1968; Bullard and Parker, 1971); hence non-conducting boundaries above and below are suitable approx. imations for this kind of model. The effect of such mathematically convenient boundary conditions is to require that the electric current circuit should be closed within the ocean itself. The currents described here would flow in addition to the mean currents flowing in the ocean which return through the upper mantle, The equations of electromagnetism in axes fixed with respect to the earth’s surface, are the “Pre-Maxwell equations” (R. Hide, private communication, 1971): aB curl E = ; curl B = pJ; div B = 0 (1)
at
The magnetic permeability is assumed to differ insignificantly from its free space value j.t 7H 0 = 47r l0 m~(Chapman and Price, 1930; Tozer, 1959). Ohm’s law can be taken in the form: ‘
J=o(E-4-UAB)
—
current is solenoidal, i.e.: d ~=0
Fig. 1. Model configuration: a plane ocean of infinite horizontal extent, finite depth, constant conductivity between insulators, permeated by vertical magnetic field F.
—
at
(2)
Provided U is specified and the boundary conditions are given, then the problem is well posed. Poisson’s equation: = div D = div ~e~ (5 q ~ / is only required as a statement of the local space charge density consistent with the electric displacements which have arisen. The vector and scalar quantities are measured here in S.I.-units so that the magnetic induction B is in Wb/m2, the electric field E in V/m, the electric current Jis in A/rn2, the electric charge density in (C/rn3) and the velocity U of the fluid medium in m.s~. The motions in this model ocean are taken to be divergence-free, and to have no dependence on the z-direction. Hence if U—= (u, u):
a
au
= 0 (6) 3’ The form of the corresponding stream function l~iis chosen to give a substantial velocity on the upper surface, and a slow return flow beneath, It satisfies the free-surface boundary condition on the top surface and also on the bottom surface where it is assumed that a weak shear in the bottom sediments will take up the small difference in velocity between the moving bottom water and the rigid crust. Such a flow could be generated by, for example, the differential heating or cooling of part of the top surface. To obtam a velocity with suitable characteristics let us suppose that it satisfies the Navier-Stokes equation in the Boussinesq approximation:
+
X
~
(7)
where, relative to the earth as frame of reference,E is
dt
the applied electric field and U A B is the electric field induced by motions in the conducting medium
0 p0 where the effects of compressibility are neglected except in the gravity term where buoyancy produces
P
R.S. Peckover, Oceanic electric currents
a body force I’. p. is the mean density, v is the kinematic viscosity (m2/s). The pressure p* is the residual after hydrostatic pressure has been subtracted. The horizontal length scale X of a typical cell is incorporated into the model by assuming periodicity in the horizontal direction normal to the roll-axis.
3. Field distributions In this section, induced electric field distributions and maximum velocities are given which are appropriate for the slow convective solutions of eq. 9. Define the Hartmann number M to be Boh/fio (Roberts, 1967), where B. is the undisturbed magnetic induction, Y is the viscosity, TJthe resistivity, and p. the mean density. Let k be the horizontal wavenumber = nn, where IZ= 1, 2,3, . . . and define the k-Hartmann number Mk by: Mk=$
(dMT+M)
(8)
ThenMk+MasM+m;andMk+kasM+O. For a steady, slow, two-dimensional circulation driven by a vertical body force of the form ro cos kX sinh ykY, the velocity U must satisfy: (9) where y is the aspect ratio h/X, X = x/X and Y = y/h. Such a body force, appropriate for differential heating, is taken to give a steady convective motion. Other driving forces which give convective turnover could easily be used instead. It is assumed in this simplified model that there is no applied electric field. Taking the curl of eq. 9, we obtain for the y-component of the vorticity {:
(
vv2_p
UBg
1
pa;
=o
(10)
The vorticity has only this one component directed normal to the vertical plane containing magnetic field and velocity vectors. Its magnitude, suitably normalized, is: { =
2 k2 <( Y - s;z;kysin kX
(11)
The corresponding 197 1):
rcI=Go
139
stream function
is (Peckover,
y_
sinh ykY sinh yk sinh yMk Y
+ (M-2) k2 k
sinh -yMk
- Y
:11
sinkX
(12)
The induced electric field Eind is horizontal and proportional in magnitude to the horizontal velocity. It is: cash ykY sinhyk
Eind
k
‘i$k
cash yMkY SiIlh
yMk
sin kX (13)
For this steady circulation the only current flowing is the induced current j (electrostatic potentials parallel to the roll are not considered here) where: j= aEind = a(UAF),
= a Fu
(14)
In eq.12 the second term (-MF2) is a viscous layer to enable the no-stress boundary condition to be satisfied. This term can be discarded if viscosity is completely neglected; if it is, no boundary condition can be applied to the (horizontal) velocity at the surface since the Navier-Stokes equation is then reduced from a second-order partial differential equation to a first-order one. In this case the solutions simplify to: $I
=+,
l
G =rik2
sin kX {Y}sinkX
(15)
Eind
The limit v + 0 is equivalent to Mk + M + m. Fig.2 shows the streamlines for the convective cell. The distance between two streamlines is inversely proportional to the velocity. Fig.3 shows the strength of the induced electric field. It is normal to the plane of the diagram, and is “out of’ the paper in the upper portion and “into” the paper in the lower portion.
R.S. Peckover, Oceanic electric currents
140
__________ ________________
Fig.2. Stream lines for the calculated flow pattern.
Since / cx uElnd, this figure also shows the induced electric current distribution, The figures are drawn for the electric field and the stream lines of eq.15. The distributions corresponding to eq.12 and 13 differ from these insignificantly. The maximum velocity is given from eq.9 to be: (16)
k(M1 — coth 7k ~ coth 7Mk) which, when viscosity is negligible, tends to: Urn
=
Um= =
—
~bo(1
—
coth 7k)
(17)
This flows horizontally near the surface. The maximum upwelling occurs for Y inviscid limit, the upwelling velocity at Y =
(
=kc~
Ui
1
02 2
~.
In the is:
sinh 7 k/2\ sinh7k
The mean electrical conductivity of sea water a
/
(18)
4. Discussion For this model, it is of interest to insert typical velocities and to consider the magnitudes of the corresponding fields and currents. Mean velocities of moving water in the open ocean have a maximum around 1.5 msec1. Surface ocean currents actually often have their maximum velocities 50—lOOm below the surface, where the North __________________________________ +
/ / I
~——-.~ ———~
/
—
Pacific Equatorial Counter Current has a maximum speed of-i msec’ (Knau~s,1961), the Atlantic Equatorial and the Gulf Stream have maximum speeds of l—1~-msec~(Neumann, 1968). Thus a velocity of 1 msec1 (i.e., ‘-~2knots) can be taken as a typical current velocity at or close to the surface. Speeds at depth are by no means negligible. The counter current below the Gulf Stream, which flows below 1500m has a velocity of 0.05—0.1 msec’. The Antarctic circumpolar current has velocities up to 0.2 msec’ at depth 3,000—5,000m (Callahan, 1971; Neumann, 1968). Velocities at the bottom of Drake Passage are estimated to be 0.6 msec1 (Neumann, 1968). West of Bermuda, Swallow (1955) has measured 0.4 msec1 at a depth of 4,000m. Upwelling times are not known. Hidaka (1954) estimated 80m/month, i.e. -‘3nWday. However, several hundred years is quoted as a typical water circulation time for bottom water (Eady, 1964). If it took 250 years to rise to the surface, this would correspond to 4 cm/day, compared with which, crustal plate motions of 4 cm/year (Whitehead, 1972), can be neglected.
\
-‘,‘
~-
/
Fig.3. Contours ofconstant electric current density (+ • “into the paper”; — = “out ~f the paper”) corresponding to Fig.2.
canbetakentobe3.3mhom~ (BullardandParker, 3. Hence magnetic of’-’~G,—l0~ the Alfvén 1971), for anda the densityfield of seawater kg/m velocity VA is ‘~—20msec~,which is low compared with the speed of sound (1,500 msec1). Indeed compressibiity can be neglected for hydromagnetic studies in the ocean. The mean ocean depth is 3 103m, the kinematic molecular viscosity of sea water is 10—6 m2 sec~,and the mean electrical resistivity r~is l0~m2 sec~.Hence the Hartmann number M = 3 106, whereas the magnetic Prandtl number (v/n) is 10—11. Taking a typical stream velocity U to be 1 msec~ then the magnetic Reynolds number is .~10_2.These parametric values justify the replacement in Ohm’s law of B by F, the (constant) main geomagnetic field. For a surface velocity of 1 msec~,and vertical magnetic field of ~ G, the maximum induced electric field is ‘~-‘2’l0~V/m, which drives a current density ‘
,
of j~ mAim2. The contours in Fig.3 are thus spaced 30 jiA/m2 apart. The return current has a maximum of ‘-‘70 #A/m2. If most of the current is concentrated
R.S. Peckover, Oceanic electric currents
in the top lOOm then the perturbation in the magnetic field is approximately 25 7. Fig.3 represents contours of electric field as well as of electric current with the translation: 1 V/m ~ 3.3 Aim2, Longuet-Higgins (1949) calculated the steady dcctric currents which result for sea flows in the earth’s magnetic field up straight channels of uniform rectangular or semi-elliptical cross-sections and conducting bottoms. He found the horizontal electric potential gradient in the water to be almost independent of vertical variation in the velocity of the water. However in his analysis the magnitude of the conductivity of the sea bed was important since part of the current path lays through it. Longuet-Higgins’ model has been borne out in estuaries by the work of Barber (1948) who found 16 mV/km at spring tides in the channel. There is renewed interest in this area. The model described in section 3 differs from this work in having no net vertically averaged current or induced electric field. Moreover the flowis rotational which enables volume space charge to accumulate (Longuet. Higgins, 1949); indeed space charge is directly propor. tional to the density of vorticity when the main magnetic field is constant. As a model of a thin-sheet ocean, Bullard and Parker (1971) looked at the horizontal (radial) flow in a circular disc. For a vertical B0 of 0.4 G and U = 1 knot they obtain an integrated steady current of 44 perturbation mA/rn for a depth 4 km. The resulting horizontal to theof magnetic field, 28 7, makes a negligible contribution to Ohm’s law. A superconducting layer located some distance below the sea floor provides the path for the return current in their model. The quasi-steady travelling of oceanic waves gives a background noise to marine magnetic measurements of the order of I or 2 For example Fraser (1966) took a horizontal velocity of ~ cx cosh kY sin kX/sinh k and found that the passage of waves with this velocity gives a magnetic field variation of a few gamma. This velocity profile is essentially the hyperbolic part of eq.15. Malkus and Stern (1952) have looked at the dcctric fields induced by an ocean current such as the Gulf Stream, flowing through a magnetic field and they established a relationship between the mean induced potential difference and the total volume of water involved in the stream. Any vertical distribu. tion of velocities could be incorporated into their ~.
141
formulation including the one used here. Stommel (1948) also considered the motion of a shallow layer over stationary water of great depth. He found that in a vertical magnetic field the electric field would be reduced in the ratio ofh 1/(h1 + h2) below the value it would have if no stationary layer were present. If the circulation of the kind described in section 3 is restricted to the top few hundred metres of the ocean then a Stommel factor must be included to take into account the integrated finite conductivity of this stationary layer. Glasko and Sveshnikov (1961) have considered two layers of finite width in which the velocities of flow were in opposite direction. This is quite a good representation of the kind of model of section 3, but the authors were concerned mainly with the jumps in the induced electric field at the vertical edges of such streams. Those effects are ignored in this paper, but could be taken into account in a more elaborate treatment. The normal oceanic daily variation is about 20 7 and this provides one yardstick against which the hydrodynamically induced variations can be measured. On the other hand this is quite close to the sensitivity limit of present instruments. For example a tilt of 1 mm, of arc in a measuring device in middle latitudes would produce an error of 107 (Bullard and Parker, 1971). The induced electric fields of a few2,mV/km, for the and model corresponding currents in ofpractice a few j.zA/m described here provide upper limits to the contribution of the vertical circulation to the electric current distribution in the oceans. Some of the dcctric current flowing horizontally near the surface can be considered as part of a set of horizontal closed current ioops, and some will return through a subterranean path for which the superconducting layer provides an idealization. The long-period long-wavelength components of the earth’s surface magnetic field provide the main source of information about the magnetic field in the earth’s core. By identifying and subtracting theeffects of oceanic motion on such signals, the coreinduced variations can be more readily understood, .
5. Jovian induced fields If higherfluid velocities (by a factor of a 100) occurred on a large scale (and this is a real possibility in the
142
R.S. Peckover, Oceanic electric currents
Acknowledgments
_____
Fig.4. Magnetic field lines distorted by motions of high magnetic Reynolds number (Rm ~‘ 1).
¶~
_______________________________________________ -
::
tion _______________________________________ Fig.5. Distribution of electric current corresponding to Fig.4 (++ = strong “into” the paper;.. . = weak “into” the paper; —— = weak “out of” the paper; g = strong “out of” the paper).
atmosphere of Jupiter (Hide, 1971) then the magnetic field would be substantially distorted. Rikitake (1966) shows the distortion of 1uced the magb ne mc e-te 14 mi m 11~y~or izontal which iS ~ ro’ y a simple sinusoidal eddy. For eudies with higher velocities the distortion in the magnetic field is greater. For extremely strong eddies, the magnetic field is cxpelled from the centre of the eddy (Weiss, 1966; Peckover and Weiss 1972). In the process, the magnetic field lines break and recombine to form closed loops, each of which gradually shrinks and finally disappears. In a true steady state no closed loops exist (cf. Rikitake, 1966 p.2’7). A non-linear solution computed using numerical finite difference methods can be used to illustrate the result. The magnetic field is concentrated into ropes, from which the flow is to some extent excluded. Strong induced fields and current concentrations develop consisting of two well separated vertical sheets with current flowing in opposite directions (see Fig.4 and 5). Before a steady state is reached, Alfv~noscillations must die away and any field loops must resistively diffuse together and annihilate. Of course if the normal steady state is one of turbulence maintained at some steady level, then closed loops would have a much longer life time. ~‘
‘+
,
p::.
I
_____
References Barber, N.F., 1948. Mon. Not. R. Astron. Soc., Geophys. Suppl., 5: 258. Bullard, E.C., 1972. Q. J. R. Astron. Soc., 13: 410. Bullard, E.C. and Parker, R.L., 1971. Electromagnetic inducN.Y., p. 695. Callahan J E 1971 J Geophys Res 76 5859 in the oceans. In: The Sea, IV. Wiley, New York, Chapman S and Price A T 1930 Philos Trans R Soc Lond.,Ser,A,229:427. Eady, E.T., 1964. In: D.R. Bates (Editor), The Planet Earth. Pergamon, 2nd Ed.; p. 150. Fraser, D.C., 1966. Geophys. J., 11: 507. Glasko, V.B. and Sveshnikov, A.G., 1961. Geomagn. Aeron., 1: 73, Hidaka, K., 1954. Trans. Am. Geophys. Union, 35: 431. Hide, R., 1971. Observatory. 91: 55. Knauss, J.A., 1961.J. Geophys. Res., 66: 143. Larsen, J.C., 1968. Geophys. J., 16: 47. Longuet-Higgins, M.S., Geophys. Suppl., 5: 1949. 285. Mon. Not. R. Astron. Soc., Longuet-Higgins, M.S., Stern, M.E. and Stommel, H., 1954. Pap. Phys. Oceanogr. Meteorol., 13 (1): 1. Malkus, W.V.R. and Stern, M.E., 1952. J. Mar. Res., 11: 97. Neumann, G., 1968. Ocean Currents. (Elsevier, New York, N.Y.). Peckover, R.S., 1971. Thesis, University of Cambridge. Peckover, R.S. and Weiss, N.O., 1972. Comp. Phys. Comm., 4:339. Rikitake, T., 1966. Electromagnetism and the Earth’s
Interior. Elsevier, Amsterdam. Roberts, P.H., 1967.An Introduction toMagnetohydrodynamics. Longmans, London. Stommel, H., 1948. J. Mar. Res., 7: 386. Swallow, J.C., 1955. Deep Sea Res., 3: 74. Tozer, D.C., 1959. Phys. Chem. Earth. 3:414. Weiss, N.O., 1966, Proc. R. Soc. Lond., Ser. A, 293: 310. Whitehead, J.A., 1972. Phys. Earth Planet. Inter., 5: 199.
OCEANIC ELECTRIC CURRENTS INDUCED BY FLUID CONVECTION
*
R.S. PECKOVER UKAEA Cuiham Laboratory, A bingdon (Great Britain)
Accepted for publication January 9, 1973
In addition to the electric potentials induced by the gyral motions in the oceans, horizontal electric fields and currents result from the exchange of water between the depths and the surface in the presence of the main geomagnetic field. In this note a simple model representing such a circulation is considered, and the spatial distribution of the correspondin~induced 2. electric calculated. A surface velocity 2 knots could These fields steadyiscurrents in the ocean could be of comparable with induce the normal electric currents up to 10— Amp/rn oceanic daily variation. Since a proportion of the electric current would return through the earth below the sea floor, this calculation provides an upper limit to this component. An indication is also given of the magnetic field distortion and associated electric currents which occur in a highly conducting (Jovian) ocean.
I. Introduction It is customary to consider the thin-sheet approximation for the oceans and to work with electric field and current distributions which are quantitiesvertically averaged through the thin sheet (Rikitake, 1966; Bullard and Parker, 1971; Bullard, 1972). In this note the effects of a simple form of vertical motion are considered. The form of the vertical distribution is considered for a convective cell in the ocean which consists of a relatively fast flowing stream on top and only such return flow beneath as continuity requires. The integrated effect of a fast flowing stream has been considered by Malkus and Stern (1952), and Glasko and Sveshnikov (1961) have examined a model of two layers of fluid with different velocities, Motions in an electrically conducting fluid permeated by a magnetic field induce electric currents which flow within the field and distort to some extent the magnetic field. The distortion of the main geomagnetic field F is negligible and this is well known (Longuet-Higgins, 1949; Larsen, 1968). However the induced electric *
This paper was presented at the I.A.G.A. Workshop on Electromagnetic Induction, held at the University of Edinburgh, 20—27 September 1972.
field and the resulting currents need not be negligible. Here the spatial distribution of fields is calculated for a simple, rather stylized model, which incorporates some features of a real ocean.
2. Model and equations The model considered here consists of a plane horizontal ocean which is of infinite horizontal cxtent, of uniform depth h metres, isotropic electrical conductivity a, and which is permeated by a vertical magnetic field of strength F. It is bounded above by a non-conducting atmosphere, and below by a nonconducting layer of bottom sediment and crustal rock. A convective circulation is postulated which is in the form of quasi-two dimensional rolls where the axis of the roll (the z-direction) has a length scale large compared with the horizontal convective length scale A (see Fig.l). The effects of sphericity can be neglected since the vertical scale of the ocean is ~-1O~ of the radius of the earth. Although a is obviously a function of temperature and salinity, and oceanic streams have temperatures different from their surrounding water, this dependence is weak, and the assumption of constant conductivity by Longuet-Higgins et al. (1954)
138
-
R.S. Peckover, Oceanic electric currents
~
~
~
~‘
/ / ///,
( / h metres
/ —
~////
/ 7/ / /77/
crustal rocks
o’ o —
/
in the of the Eq.presence 1 and 2 can be magnetic combinedinduction together B. into the
/ /7/.
hydromagnetic induction equation:
plane ocean infinite extent
~B —
o— const.
F
7/ / /777
curl (UAB)
V B
= i~
(3)
1is the resistivity. where (~ia) With~ithese equations, it follows that the electric 4
1V
is followed here. The Coriolis force is ignored since no motions are considered in the z-direction. At British latitudes the horizontal component is less than half the vertical component. In fact in the linearized case the solutions for horizontal and vertical components can easily be superimposed. Here only a vertical component is considered. The electrical conductivity of the atmosphere is a factor of 1012 less than that of the ocean, and the crustal rocks are a factor of 10~less conductive than the ocean (Larsen, 1968; Bullard and Parker, 1971); hence non-conducting boundaries above and below are suitable approx. imations for this kind of model. The effect of such mathematically convenient boundary conditions is to require that the electric current circuit should be closed within the ocean itself. The currents described here would flow in addition to the mean currents flowing in the ocean which return through the upper mantle, The equations of electromagnetism in axes fixed with respect to the earth’s surface, are the “Pre-Maxwell equations” (R. Hide, private communication, 1971): aB curl E = ; curl B = pJ; div B = 0 (1)
at
The magnetic permeability is assumed to differ insignificantly from its free space value j.t 7H 0 = 47r l0 m~(Chapman and Price, 1930; Tozer, 1959). Ohm’s law can be taken in the form: ‘
J=o(E-4-UAB)
—
current is solenoidal, i.e.: d ~=0
Fig. 1. Model configuration: a plane ocean of infinite horizontal extent, finite depth, constant conductivity between insulators, permeated by vertical magnetic field F.
—
at
(2)
Provided U is specified and the boundary conditions are given, then the problem is well posed. Poisson’s equation: = div D = div ~e~ (5 q ~ / is only required as a statement of the local space charge density consistent with the electric displacements which have arisen. The vector and scalar quantities are measured here in S.I.-units so that the magnetic induction B is in Wb/m2, the electric field E in V/m, the electric current Jis in A/rn2, the electric charge density in (C/rn3) and the velocity U of the fluid medium in m.s~. The motions in this model ocean are taken to be divergence-free, and to have no dependence on the z-direction. Hence if U—= (u, u):
a
au
= 0 (6) 3’ The form of the corresponding stream function l~iis chosen to give a substantial velocity on the upper surface, and a slow return flow beneath, It satisfies the free-surface boundary condition on the top surface and also on the bottom surface where it is assumed that a weak shear in the bottom sediments will take up the small difference in velocity between the moving bottom water and the rigid crust. Such a flow could be generated by, for example, the differential heating or cooling of part of the top surface. To obtam a velocity with suitable characteristics let us suppose that it satisfies the Navier-Stokes equation in the Boussinesq approximation:
+
X
~
(7)
where, relative to the earth as frame of reference,E is
dt
the applied electric field and U A B is the electric field induced by motions in the conducting medium
0 p0 where the effects of compressibility are neglected except in the gravity term where buoyancy produces
P
R.S. Peckover, Oceanic electric currents
a body force I’. p. is the mean density, v is the kinematic viscosity (m2/s). The pressure p* is the residual after hydrostatic pressure has been subtracted. The horizontal length scale X of a typical cell is incorporated into the model by assuming periodicity in the horizontal direction normal to the roll-axis.
3. Field distributions In this section, induced electric field distributions and maximum velocities are given which are appropriate for the slow convective solutions of eq. 9. Define the Hartmann number M to be Boh/fio (Roberts, 1967), where B. is the undisturbed magnetic induction, Y is the viscosity, TJthe resistivity, and p. the mean density. Let k be the horizontal wavenumber = nn, where IZ= 1, 2,3, . . . and define the k-Hartmann number Mk by: Mk=$
(dMT+M)
(8)
ThenMk+MasM+m;andMk+kasM+O. For a steady, slow, two-dimensional circulation driven by a vertical body force of the form ro cos kX sinh ykY, the velocity U must satisfy: (9) where y is the aspect ratio h/X, X = x/X and Y = y/h. Such a body force, appropriate for differential heating, is taken to give a steady convective motion. Other driving forces which give convective turnover could easily be used instead. It is assumed in this simplified model that there is no applied electric field. Taking the curl of eq. 9, we obtain for the y-component of the vorticity {:
(
vv2_p
UBg
1
pa;
=o
(10)
The vorticity has only this one component directed normal to the vertical plane containing magnetic field and velocity vectors. Its magnitude, suitably normalized, is: { =
2 k2 <( Y - s;z;kysin kX
(11)
The corresponding 197 1):
rcI=Go
139
stream function
is (Peckover,
y_
sinh ykY sinh yk sinh yMk Y
+ (M-2) k2 k
sinh -yMk
- Y
:11
sinkX
(12)
The induced electric field Eind is horizontal and proportional in magnitude to the horizontal velocity. It is: cash ykY sinhyk
Eind
k
‘i$k
cash yMkY SiIlh
yMk
sin kX (13)
For this steady circulation the only current flowing is the induced current j (electrostatic potentials parallel to the roll are not considered here) where: j= aEind = a(UAF),
= a Fu
(14)
In eq.12 the second term (-MF2) is a viscous layer to enable the no-stress boundary condition to be satisfied. This term can be discarded if viscosity is completely neglected; if it is, no boundary condition can be applied to the (horizontal) velocity at the surface since the Navier-Stokes equation is then reduced from a second-order partial differential equation to a first-order one. In this case the solutions simplify to: $I
=+,
l
G =rik2
sin kX {Y}sinkX
(15)
Eind
The limit v + 0 is equivalent to Mk + M + m. Fig.2 shows the streamlines for the convective cell. The distance between two streamlines is inversely proportional to the velocity. Fig.3 shows the strength of the induced electric field. It is normal to the plane of the diagram, and is “out of’ the paper in the upper portion and “into” the paper in the lower portion.
R.S. Peckover, Oceanic electric currents
140
__________ ________________
Fig.2. Stream lines for the calculated flow pattern.
Since / cx uElnd, this figure also shows the induced electric current distribution, The figures are drawn for the electric field and the stream lines of eq.15. The distributions corresponding to eq.12 and 13 differ from these insignificantly. The maximum velocity is given from eq.9 to be: (16)
k(M1 — coth 7k ~ coth 7Mk) which, when viscosity is negligible, tends to: Urn
=
Um= =
—
~bo(1
—
coth 7k)
(17)
This flows horizontally near the surface. The maximum upwelling occurs for Y inviscid limit, the upwelling velocity at Y =
(
=kc~
Ui
1
02 2
~.
In the is:
sinh 7 k/2\ sinh7k
The mean electrical conductivity of sea water a
/
(18)
4. Discussion For this model, it is of interest to insert typical velocities and to consider the magnitudes of the corresponding fields and currents. Mean velocities of moving water in the open ocean have a maximum around 1.5 msec1. Surface ocean currents actually often have their maximum velocities 50—lOOm below the surface, where the North __________________________________ +
/ / I
~——-.~ ———~
/
—
Pacific Equatorial Counter Current has a maximum speed of-i msec’ (Knau~s,1961), the Atlantic Equatorial and the Gulf Stream have maximum speeds of l—1~-msec~(Neumann, 1968). Thus a velocity of 1 msec1 (i.e., ‘-~2knots) can be taken as a typical current velocity at or close to the surface. Speeds at depth are by no means negligible. The counter current below the Gulf Stream, which flows below 1500m has a velocity of 0.05—0.1 msec’. The Antarctic circumpolar current has velocities up to 0.2 msec’ at depth 3,000—5,000m (Callahan, 1971; Neumann, 1968). Velocities at the bottom of Drake Passage are estimated to be 0.6 msec1 (Neumann, 1968). West of Bermuda, Swallow (1955) has measured 0.4 msec1 at a depth of 4,000m. Upwelling times are not known. Hidaka (1954) estimated 80m/month, i.e. -‘3nWday. However, several hundred years is quoted as a typical water circulation time for bottom water (Eady, 1964). If it took 250 years to rise to the surface, this would correspond to 4 cm/day, compared with which, crustal plate motions of 4 cm/year (Whitehead, 1972), can be neglected.
\
-‘,‘
~-
/
Fig.3. Contours ofconstant electric current density (+ • “into the paper”; — = “out ~f the paper”) corresponding to Fig.2.
canbetakentobe3.3mhom~ (BullardandParker, 3. Hence magnetic of’-’~G,—l0~ the Alfvén 1971), for anda the densityfield of seawater kg/m velocity VA is ‘~—20msec~,which is low compared with the speed of sound (1,500 msec1). Indeed compressibiity can be neglected for hydromagnetic studies in the ocean. The mean ocean depth is 3 103m, the kinematic molecular viscosity of sea water is 10—6 m2 sec~,and the mean electrical resistivity r~is l0~m2 sec~.Hence the Hartmann number M = 3 106, whereas the magnetic Prandtl number (v/n) is 10—11. Taking a typical stream velocity U to be 1 msec~ then the magnetic Reynolds number is .~10_2.These parametric values justify the replacement in Ohm’s law of B by F, the (constant) main geomagnetic field. For a surface velocity of 1 msec~,and vertical magnetic field of ~ G, the maximum induced electric field is ‘~-‘2’l0~V/m, which drives a current density ‘
,
of j~ mAim2. The contours in Fig.3 are thus spaced 30 jiA/m2 apart. The return current has a maximum of ‘-‘70 #A/m2. If most of the current is concentrated
R.S. Peckover, Oceanic electric currents
in the top lOOm then the perturbation in the magnetic field is approximately 25 7. Fig.3 represents contours of electric field as well as of electric current with the translation: 1 V/m ~ 3.3 Aim2, Longuet-Higgins (1949) calculated the steady dcctric currents which result for sea flows in the earth’s magnetic field up straight channels of uniform rectangular or semi-elliptical cross-sections and conducting bottoms. He found the horizontal electric potential gradient in the water to be almost independent of vertical variation in the velocity of the water. However in his analysis the magnitude of the conductivity of the sea bed was important since part of the current path lays through it. Longuet-Higgins’ model has been borne out in estuaries by the work of Barber (1948) who found 16 mV/km at spring tides in the channel. There is renewed interest in this area. The model described in section 3 differs from this work in having no net vertically averaged current or induced electric field. Moreover the flowis rotational which enables volume space charge to accumulate (Longuet. Higgins, 1949); indeed space charge is directly propor. tional to the density of vorticity when the main magnetic field is constant. As a model of a thin-sheet ocean, Bullard and Parker (1971) looked at the horizontal (radial) flow in a circular disc. For a vertical B0 of 0.4 G and U = 1 knot they obtain an integrated steady current of 44 perturbation mA/rn for a depth 4 km. The resulting horizontal to theof magnetic field, 28 7, makes a negligible contribution to Ohm’s law. A superconducting layer located some distance below the sea floor provides the path for the return current in their model. The quasi-steady travelling of oceanic waves gives a background noise to marine magnetic measurements of the order of I or 2 For example Fraser (1966) took a horizontal velocity of ~ cx cosh kY sin kX/sinh k and found that the passage of waves with this velocity gives a magnetic field variation of a few gamma. This velocity profile is essentially the hyperbolic part of eq.15. Malkus and Stern (1952) have looked at the dcctric fields induced by an ocean current such as the Gulf Stream, flowing through a magnetic field and they established a relationship between the mean induced potential difference and the total volume of water involved in the stream. Any vertical distribu. tion of velocities could be incorporated into their ~.
141
formulation including the one used here. Stommel (1948) also considered the motion of a shallow layer over stationary water of great depth. He found that in a vertical magnetic field the electric field would be reduced in the ratio ofh 1/(h1 + h2) below the value it would have if no stationary layer were present. If the circulation of the kind described in section 3 is restricted to the top few hundred metres of the ocean then a Stommel factor must be included to take into account the integrated finite conductivity of this stationary layer. Glasko and Sveshnikov (1961) have considered two layers of finite width in which the velocities of flow were in opposite direction. This is quite a good representation of the kind of model of section 3, but the authors were concerned mainly with the jumps in the induced electric field at the vertical edges of such streams. Those effects are ignored in this paper, but could be taken into account in a more elaborate treatment. The normal oceanic daily variation is about 20 7 and this provides one yardstick against which the hydrodynamically induced variations can be measured. On the other hand this is quite close to the sensitivity limit of present instruments. For example a tilt of 1 mm, of arc in a measuring device in middle latitudes would produce an error of 107 (Bullard and Parker, 1971). The induced electric fields of a few2,mV/km, for the and model corresponding currents in ofpractice a few j.zA/m described here provide upper limits to the contribution of the vertical circulation to the electric current distribution in the oceans. Some of the dcctric current flowing horizontally near the surface can be considered as part of a set of horizontal closed current ioops, and some will return through a subterranean path for which the superconducting layer provides an idealization. The long-period long-wavelength components of the earth’s surface magnetic field provide the main source of information about the magnetic field in the earth’s core. By identifying and subtracting theeffects of oceanic motion on such signals, the coreinduced variations can be more readily understood, .
5. Jovian induced fields If higherfluid velocities (by a factor of a 100) occurred on a large scale (and this is a real possibility in the
142
R.S. Peckover, Oceanic electric currents
Acknowledgments
_____
Fig.4. Magnetic field lines distorted by motions of high magnetic Reynolds number (Rm ~‘ 1).
¶~
_______________________________________________ -
::
tion _______________________________________ Fig.5. Distribution of electric current corresponding to Fig.4 (++ = strong “into” the paper;.. . = weak “into” the paper; —— = weak “out of” the paper; g = strong “out of” the paper).
atmosphere of Jupiter (Hide, 1971) then the magnetic field would be substantially distorted. Rikitake (1966) shows the distortion of 1uced the magb ne mc e-te 14 mi m 11~y~or izontal which iS ~ ro’ y a simple sinusoidal eddy. For eudies with higher velocities the distortion in the magnetic field is greater. For extremely strong eddies, the magnetic field is cxpelled from the centre of the eddy (Weiss, 1966; Peckover and Weiss 1972). In the process, the magnetic field lines break and recombine to form closed loops, each of which gradually shrinks and finally disappears. In a true steady state no closed loops exist (cf. Rikitake, 1966 p.2’7). A non-linear solution computed using numerical finite difference methods can be used to illustrate the result. The magnetic field is concentrated into ropes, from which the flow is to some extent excluded. Strong induced fields and current concentrations develop consisting of two well separated vertical sheets with current flowing in opposite directions (see Fig.4 and 5). Before a steady state is reached, Alfv~noscillations must die away and any field loops must resistively diffuse together and annihilate. Of course if the normal steady state is one of turbulence maintained at some steady level, then closed loops would have a much longer life time. ~‘
‘+
,
p::.
I
_____
References Barber, N.F., 1948. Mon. Not. R. Astron. Soc., Geophys. Suppl., 5: 258. Bullard, E.C., 1972. Q. J. R. Astron. Soc., 13: 410. Bullard, E.C. and Parker, R.L., 1971. Electromagnetic inducN.Y., p. 695. Callahan J E 1971 J Geophys Res 76 5859 in the oceans. In: The Sea, IV. Wiley, New York, Chapman S and Price A T 1930 Philos Trans R Soc Lond.,Ser,A,229:427. Eady, E.T., 1964. In: D.R. Bates (Editor), The Planet Earth. Pergamon, 2nd Ed.; p. 150. Fraser, D.C., 1966. Geophys. J., 11: 507. Glasko, V.B. and Sveshnikov, A.G., 1961. Geomagn. Aeron., 1: 73, Hidaka, K., 1954. Trans. Am. Geophys. Union, 35: 431. Hide, R., 1971. Observatory. 91: 55. Knauss, J.A., 1961.J. Geophys. Res., 66: 143. Larsen, J.C., 1968. Geophys. J., 16: 47. Longuet-Higgins, M.S., Geophys. Suppl., 5: 1949. 285. Mon. Not. R. Astron. Soc., Longuet-Higgins, M.S., Stern, M.E. and Stommel, H., 1954. Pap. Phys. Oceanogr. Meteorol., 13 (1): 1. Malkus, W.V.R. and Stern, M.E., 1952. J. Mar. Res., 11: 97. Neumann, G., 1968. Ocean Currents. (Elsevier, New York, N.Y.). Peckover, R.S., 1971. Thesis, University of Cambridge. Peckover, R.S. and Weiss, N.O., 1972. Comp. Phys. Comm., 4:339. Rikitake, T., 1966. Electromagnetism and the Earth’s
Interior. Elsevier, Amsterdam. Roberts, P.H., 1967.An Introduction toMagnetohydrodynamics. Longmans, London. Stommel, H., 1948. J. Mar. Res., 7: 386. Swallow, J.C., 1955. Deep Sea Res., 3: 74. Tozer, D.C., 1959. Phys. Chem. Earth. 3:414. Weiss, N.O., 1966, Proc. R. Soc. Lond., Ser. A, 293: 310. Whitehead, J.A., 1972. Phys. Earth Planet. Inter., 5: 199.